In NSW you are entitled to lodge a claim against the "Nominal Defendant" (the Nominal Defendant is a scheme run by the State government for exactly this situation).
However, strict time limits apply. You need to report the accident to police within 28 days and lodge a Personal Injury Claim Form (available on the Motor Accident Authority of NSW website) within 6 months of the date of the accident. When lodging a claim upon the Nominal Defendant you are also required to conduct "due search and enquiry" in an attempt to establish the registration details of the unknown vehicle.
It depends on how the injury or psychological injury occurred. All States in Australia have varying systems of compensation - the process and the compensation available vary greatly.
However, most systems (those in NSW most certainly) allow claims which relate to psychological injury.
In NSW a person injured in a motor vehicle accident is required to report the accident to police within 28 days.